algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
quantum mechanical system, quantum probability
interacting field quantization
The concept of BV-operator has two different but (somewhat subtly) related meanings:
In perturbative quantum field theory (discussed below) the BV-operator or BV-Laplacian (Batalin-Vilkovisky 81) is essentially the difference between the action of the BV differential on the algebra of observables before and after quantization of the free field theory around which the perturbative quantization is considered; it is a quantum correction $i \hbar \Delta_{BV}$ (of order of Planck's constant $\hbar$) to the BV differential $s_{BV} = \{-S',-.\}$. Where the condition $\left(\{-S',-\}\right)^2 = 0$ for the BV-differential to be a differential is called the “master equation” in BV-BRST theory, the quantum corrected version $\left( \{-S',-\} + i \hbar \Delta\right)^2 = 0$ is called the quantum master equation.
In higher algebra, under the identification of a BV-algebra with the chain homology of a E2-algebra, the BV-operator corresponds to the operation of rotating a little disk around.
For the relation between the two see relation between BV and BD.
In perturbative quantum field theory the BV-operator $i \hbar \Delta$ may be understood intuitively as reflecting the contribution of the Gaussian measure in the path integral of the free field theory around which the perturbative quantization takes place. This intuition may be made precise for finite-dimensional toy path integrals. This we discuss in:
In the rigorous construction of relativistic perturbative quantum field theory via causal perturbation theory/perturbative AQFT there is a rigorous incarnation of the BV-operator (Fredenhagen-Rejzner 11b, section 2, Rejzner 11, section 5.1.2): The would-be path integral is reflected in the perturbative S-matrix, hence in the time ordered products, and the BV-operator on regular polynomial observables is the difference between the classical BV-differential and its conjugation into the time-ordered products (def. 2 below). This we discuss in
If $X$ is a finite dimensional closed oriented smooth manifold, then integration of differential forms of top degree over $X$ may be identified with sending such differential forms to their image in the de Rham cohomology of $X$.
This may be slightly reformulated: Fixing a volume form $\mu$ on $X$ it induces by contraction with vector fields degreewise a linear isomorphism
between the spaces of differential n-forms on $X$ and the space of multivector fields on $X$ of degree $n-dim(X)$. Under this isomorphism the de Rham differential induces a differential on multivector fields:
This $\Delta_{BV}$ is the BV-operator in this simple situation.
The above statement about integration now translates into saying that for $f$ any smooth function on $X$, then its integration of differential forms $\int f \mu$ may be identified with the image of $f$ in the chain homology of the BV-operator $\Delta_{BV}$.
If one thinks of $X$ as a space of field configurations and of $f = \exp(i \hbar S)$ as an exponentiated action functional, the one may think of this integral $\int \exp(i \hbar S) \mu$ as the finite-dimensional toy version of a path integral.
While this is in general not defined in the actual non-finite dimensional situations in field theory, the above re-formulation in terms of the chain homology of a BV-operator does make sense whenever an appropriate kind of differential is given. One may then try to axiomatize which chain complexes qualify as BV-complex and try to interpret their chain homology as computing perturbative path integrals.
For more on this perspective see at BV-BRST formalism the section Quantum BV as homological (path-)integration
Recall the following context
Let $(E,\mathbf{L})$ be a free Lagrangian field theory admitting a gauge fixing, and let $\mathbf{L}' - \mathbf{L}'_{BRST}$ be its BV-BRST Lagrangian density after gauge fixing (this def.), so that the gauge-fixed local BV-BRST differential is given by the local antibracket as
The corresponding global BV-BRST differential on regular polynomial observables is (this def.)
By definition of gauge fixing (this def.), the Euler-Lagrange equation of motion for $\mathbf{L}'$ are Green hyperbolic and hence have a causal propagator $\Deta = \Delta_+ - \Delta_-$ and admit a compatible Wightman propagator $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ and the corresponding Feynman propagator $\Delta_F \coloneqq \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$. The star products with respect to these (this def.) on regular polynomial observables
are, respectively, the Wick algebra product (operator product, see this def.)
and the time-ordered product (see again this def.)
Since the Feynman propagator is symmetric (this prop.), the latter time ordered product on regular polynomial observables is isomorphic (via this prop.) to the pointwise product, via
as
(this prop.).
Let $(E,\mathbf{L})$ be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (this def.) on a graded BV-BRST field bundle $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.
Then the time-ordered global antibracket on regular polynomial observables
is the conjugation of the global antibracket (this def.) by the time-ordering operator $\mathcal{T}$ (from this prop.):
hence
(Fredenhagen-Rejzner 11, (27), Rejzner 11, (5.14))
(time-ordered antibracket with gauge fixed action functional)
Let $(E,\mathbf{L})$ be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (this def.) on a graded BV-BRST field bundle $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.
Then the time-ordered antibracket (def. 1) with the gauge fixed BV-action functional $-S'$ (def. \ref{ComplexBVBRSTGlobal}) equals the conjugation of the global BV-differential with the isomorphism $\mathcal{T}$ from the pointwise to the time-ordered product of observables (from this prop.)
hence
By the assumption that $(E,\mathbf{L})$ is a free field theory its Euler-Lagrange equations are linear in the fields, and hence $S'$ is quadratic in the fields. This means that
where the second term on the right is independent of the fields, and hence that
This implies the claim:
(BV-operator for gauge fixed free Lagrangian field theory)
Let $(E,\mathbf{L})$ be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (this def.) on a graded BV-BRST field bundle $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ and with corresponding gauge-fixed global BV-BRST differential on graded regular polynomial observables
(this def.).
Then the corresponding BV-operator
on regular polynomial observables is, up to a factor of $i \hbar$, the difference between the free component $\{-S',-\}$ of the gauge fixed global BV differential its time-ordered version (def. 1)
hence
(BV-operator in components)
If the field bundles of all fields, ghost fields and auxiliary fields are trivial vector bundles, with field/ghost-field/auxiliary-field coordinates collectively denoted $(\phi^A)$ then the BV-operator $\Delta_{BV}$ from prop. 2 is given explicitly by
Since this formula exhibits a graded Laplace operator, the BV-operator is also called the BV-Laplace operator or BV-Laplacian, for short.
(Fredenhagen-Rejzner 11, (29), Rejzner 11, (5.20))
By prop. 1 we have equivalently
and by this example the second term on the right is
With this we compute as follows:
Here we used
under the first brace that by assumption of a free field theory, $\{-S',-\}$ is linear in the fields, so that the first commutator with the Feynman propagator is independent of the fields, and hence all the higher commutators vanish;
under the second brace that the Feynman propagator is $+i$ times the Green function for the Green hyperbolic Euler-Lagrange equations of motion (this cor.).
(global antibracket exhibits failure of BV-operator to be a derivation)
Let $(E,\mathbf{L})$ be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (this def.) on a graded BV-BRST field bundle $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$
The BV-operator $\Delta_{BV}$ (def. 2) and the global antibracket $\{-,-\}$ (this def.) satisfy for all polynomial observables (this def.) $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ]$ the relation
for $(-) \cdot (-)$ the pointwise product of observables (def. \ref{Observable}).
Moreover, it commutes on regular polynomial observables with the time-ordering operator $\mathcal{T}$ (this prop.)
and hence satisfies the analogue of relation (5) also for the time-ordered antibracket $\{-,-\}_{\mathcal{T}}$ (def. 1) and the time-ordered product $\star_F$ on regular polynomial observables
(e.g. Henneaux-Teitelboim 92, (15.105d))
With prop. 2 the first statement is a graded version of the analogous relation for an ordinary Laplace operator $\Delta \coloneqq g^{a b} \partial_a \partial_b$ acting on smooth functions on Cartesian space, which on smooth functions $f,g$ satisfies
by the product law for differentiation, where now $\nabla f \coloneqq (g^{a b} \partial_b f)$ is the gradient and $(v,w) \coloneqq g_{a b} v^a w b$ the inner product. Here one just needs to carefully record the relative signs that appear.
That the BV-operator commutes with the time-ordering operator is clear from the fact that both of these are given by partial functional derivatives with constant coefficients. This immediately implies the last statement from the first.
(BV-operator on time-ordered exponentials)
Let $(E,\mathbf{L})$ be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (this def.) on a graded BV-BRST field bundle $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.
Let moreover $V \in PolyObs(E_{\text{BV-BRST}})_{reg, deg = 0}[ [\hbar] ]$ be a regular polynomial observable (def. \ref{PolynomialObservables}) of degree zero. Then the application of the BV-operator $\Delta_{BV}$ (def. 2) to the time-ordered exponential $\exp_{\mathcal{T}}(V)$ (this example) is the time-ordered product of the time-ordered exponential with the sum of $\Delta_{BV}(V)$ and the global time-ordered antibracket $\tfrac{1}{2}\{V,V\}_{\mathcal{T}}$ (def. 1) of $V$ with itself:
By prop. 3 $\Delta_{BV}$ acts as a derivation on the time-ordered product up to a correction given by the antibracket of the two factors. This yields the result by the usual combinatorics of exponentials.
A special case of the general occurence of the BV-operator is the following important property of on-shell time-ordered products:
Let $(E,\mathbf{L})$ be a free Lagrangian field theory (this def.) with gauge fixed BV-BRST Lagrangian density $-\mathbf{L}' + \mathbf{L}'_{BRST}$ (this def.) on a graded BV-BRST field bundle $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.
Let
be an off-shell regular polynomial observable which is linear in the antifield field observables $\mathbf{\Phi}^\ddagger$. Then
This is called the Schwinger-Dyson equation.
The proof below is due to (Rejzner 16, remark 7.7), following the informal traditional argument (Henneaux-Teitelboim 92, (15.108b)).
Applying the inverse time-ordering map $\mathcal{T}^{-1}$ (this prop.) to equation (3) applied to $A$ yields
where we have identified the terms under the braces by 1) the component expression for the BV-differential $\{-S',-\}$ from this prop, 2) prop. 2 and 3) prop. 1.
The last term is manifestly in the image of the BV-differential $\{-S',-\}$ and hence vanishes when passing to on-shell observables along the isomorphism (this equation)
(by this example.).
The same argument with the replacement $\mathcal{T} \leftrightarrow \mathcal{T}^{-1}$ throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the $\hbar$-term reversed).
(“Schwinger-Dyson operator”)
The proof of the Schwinger-Dyson equation in prop. 4 shows that, up to time-ordering, the Schwinger-Dyson equation is the on-shell vanishing of the “quantized” BV-differential (3)
where the BV-operator is the quantum correction of order $\hbar$. Therefore this is also called the Schwinger-Dyson operator (Henneaux-Teitelboim 92, (15.111)).
(distributional Schwinger-Dyson equation)
Often the Schwinger-Dyson equation (prop. 4) is displayed before spacetime-smearing of field observables in terms of operator products of operator-valued distributions, taking the observable $A$ in (6) to be
This choice makes (7) become the distributional Schwinger-Dyson equation
(e.q. Dermisek 09).
In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then
Since by the principle of extremal action (this prop.) the equation
is the Euler-Lagrange equation of motion (for the classical field theory) “at $x_0$”, this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for time-ordered quantum expectation values, as long as all other observables are evaluated away from $x_0$; while if observables do coincide at $x_0$ then there is a correction measured by the BV-operator.
(the “quantum shell”)
Beware that, superficially, it might seem that in equation (8) not only the term $\{-S',\mathcal{T}^{-1}(A)\}$ on the right vanishes on-shell, but also the term $\mathcal{T}^{-1}\left\{ -S', A\right\}$ on the left, since the latter is the image under the linear map $\mathcal{T}^{-1}$ of an observable that vanishes on-shell.
To sort this out, notice that the isomorphism (9) tells us that the observables that vanish when passing from off-shell to on-shell observables are precisely those in the ideal generated by the image of $\{-S',(-)\}$. But while $\mathcal{T}^{-1}$ is an isomorphism on (regular off-shell observables), it need not (and in general does not) preserve this ideal! Hence $\mathcal{T}^{-1}(\{-S',A\})$ need not (and in general is not) an element of that ideal, and this is why it remains when passing to the algebra of on-shell observables and thus makes its crucial appearance in the Schwinger-Dyson equation.
Passing from free field theory to perturbative quantization of interacting field theory, the above BV-operator of the underying free field theory appears as a quantum correction to the “classical master equation” which expresses the nilpotency of the BV-differential. This “master equation” corrected by the BV-operator is called the quantum master equation. See this prop..
The concept originates with
Traditional review includes
The understanding of the BV-operator in the rigorous formulation of relativistic perturbative quantum field theory via causal perturbation theory/perturbative AQFT is due to
Klaus Fredenhagen, Katarzyna Rejzner, section 2 of Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. (2013) 317: 697 (arXiv:1110.5232, doi:10.1007/s00220-012-1601-1)
Katarzyna Rejzner, section 5.1.2 of Batalin-Vilkovisky formalism in locally covariant field theory (arXiv:1111.5130)
surveyed in
See at BV-formalism for more references.